BLACK HOLE THERMODYNAMICS The horizon area theorem 1970: Stephen Hawking uses the theory of general relativity to derive the so-called

Horizon area theorem

The total horizon area in a closed system containing black holes never decreases. It can only increase or stay the same.

[Stephen Hawking] Analogy between the area theorem and the 2nd law of thermodynamic

Shortly after Stephen Hawking Formulated the area theorem, Jacob Beckenstein, at the time a graduate student at Princeton, noticed the analogy between the area theorem and the 2nd law of thermodynamics: [Jacob Beckenstein] The total area of a closed system never decreases.

Entropy: logarithm of the number of ways you can relocate the atoms and molecules of a system without changing the overall properties of the system. Example of entropy: toys in a playroom (Thorne, pg. 424) Extremely orderly: 20 toys on 1 tile This playroom floor has 100 tiles, on which the kids can arrange 20 different toys. Parents prefer the toys to be kept in an extremely orderly configuration, with all the toys piled on one tile in one corner, as shown. There is only one such arrangement; the entropy of this configuration is thus the Number of equivalent rearrangements = 1; logarithm of 1, which is zero. entropy = 0.

[This and next two slides courtesy of D. Watson] Entropy in a playroom (continued)

Orderly: 20 toys on 10 tiles Parents might even accept this somewhat less orderly configuration: 20 different toys on 10 specific tiles. But there are lots of different equivalent arrangements (e.g. swapping the positions of two toys on different tiles produces a different arrangement that’s still acceptable): 1020 of them, in Number of equivalent rearrangements = 1020; fact, for an entropy value of “entropy” = 20. log(1020) = 20. Entropy in a playroom (continued)

Disorderly: 20 toys on 100 tiles Of course, when the kids are done playing, the floor looks like this: 20 different toys spread randomly over 100 tiles. There are 10020 = 1040 different ways to do that; the entropy is 40. And kids are like natural physical processes. Through their agency the room will not get less random; the entropy of the room full of toys and Number of equivalent kids never decreases by itself. rearrangements = 1040; “entropy” = 40. Hawking-Beckenstein debate (1972)

Hawking: the analogy between the area theorem and the 2nd law of thermodynamics is just a matter of coincidence.

Beckenstein: I am not convinced. Nowhere in nature the 2nd law of thermodynamics is violated. Why black holes would be an exception? I believe that the area of black holes is acutally a manifestation of their entropy.

Wheeler (Beckenstein’s thesis advisor) to Beckenstein: Your idea is just so crazy that it might actually be true..

Hawking: if a BH has entropy, it must have a temperature, and if it has a temperature it must radiate like a blackbody. But if nothing can escape from a BH, how can it radiate? 1974: end of the debate and discovery of the Hawking radiation

1974: Hawking admits that Beckenstein was right: Black Holes have entropy, which is proportional to their area. However, this no longer constitutes a problem, because Hawking discovers a mechanism by which black holes can radiate. This mechanism involves quantum-mechanical processes near the BH horizon. According to quantum mechanics, vacuum is no longer an “empty” state as in classical physics. Instead, it is a place where “virtual particles” are constantly created and destroyed. The creation of a couple of particle-antiparticle pair of energy D E is allowed in quantum mechanics as long as their lifetime D t satisfies the Heisenberg uncertainty principle, that can also be written as DEDt ~ h Notice how, the larger the energy of the particles, the smaller can their survival† times be. These quantum-mechanical processes† of particle-antiparticles creation and destruction are also known as “vacuum fluctuations”. † Virtual particles in quantum mechanics

Examples of particle-antiparticle pairs made from the vacuum:

Electron Green photon Human

Positron Green photon Anti-human

6x10-22 seconds 1x10-16 seconds 2x10-53 seconds

Energy = Energy = Energy = 1.6x10-6 erg 8x10-12 erg 4.5x1025 erg

[slide courtesy of D. Watson] HOW DOES THE HAWKING EFFECT WORK? Creation of “virtual pairs” “VACUUM FLUCTUATIONS” of particles

In the very short time that virtual pairs can exist...

Tidal forces pull them apart

Makes some of them real

One falls in, one flies away

Black hole evaporates

[slide courtesy of M. Begelman] Black hole evaporation Hawking radiation is emitted more efficiently if the tidal forces near the horizon are stronger, i.e. for smaller mass black holes. The temperature of a black hole is inversely proportional to its mass; therefore higher temperature means smaller mass. The evaporation time is proportional to (mass)3. Here are some typical evaporation times:

9 94 10 Msun BH: 10 yr 67 2 Msun BH: 10 yr 108 g BH: 1 sec

At the final stage, when the BH temperature reaches trillions of K, the BH explodes. [simulation of BH explosion by A. Hamilton] Hawking’s latest discovery (2004): information is not completely lost when falling into a BH into a black hole

If you jump into a BH, your mass and energy will be returned to our Universe, but in a mangled form, which contains information about what you were like, but in an unrecognizable state